The \(\mathsf {S}_n\)  term:
It is worth noting our discussions of this \(\mathsf {S}_n\)  term is substantially different than that of [1]}. In particular, while the integral of the \(j\) th summand of \(\mathsf {S}_n\)  has an explicit form due to [2]}, but for our purposes of obtaining moment bounds, knowing this explicit form is unnecessary. In contrast to our purposes, the desired computation of the predicted value \(\mathbb {E}[ \left\langle \mathsf {S}_n \rho ,X_{n + 1} \right\rangle ^2]\)  in [1]} gives them an extra smoothing property which can take advantage of the explicit form of \(\mathsf {S}_n\) . Indeed, an earlier draft of this paper uses analogous proofs methods of this \(\mathsf {S}_n\)  term as the authors and we arrive at the same rate in (REF ).
By triangle inequality,
\(\vert \vert \mathsf {S}_n\vert \vert &\le \frac{1}{2\pi } \sum _{j = 1}^{k_n} \int _{\mathcal {B}_j} \vert \vert  (z - \Gamma )^{-1} (\Gamma _n - \Gamma ) \vert \vert  \,\vert \vert (z - \Gamma )^{-1}\vert \vert  \,\mathrm {d}z\) 
